+− Contents

1 Purpose

F04CCF computes the solution to a complex system of linear equations AX=B, where A is an n by n tridiagonal matrix and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

3 Description

The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is unit lower triangular with at most one nonzero subdiagonal element, and U is an upper triangular band matrix with two superdiagonals. The factored form of A is then used to solve the system of equations AX=B.

Note that the equations ATX=B may be solved by interchanging the order of the arguments DU and DL.

On exit: if IFAIL≥0, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row IPIVi. IPIVi will always be either i or i+1; IPIVi=i indicates a row interchange was not required.

On entry: the first dimension of the array B as declared in the (sub)program from which F04CCF is called.

Constraint:
LDB≥max1,N.

10: RCOND – REAL (KIND=nag_wp)Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as RCOND=1/A1A-11.

11: ERRBND – REAL (KIND=nag_wp)Output

On exit: if IFAIL=0 or N+1, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1≤ERRBND, where x^ is a column of the computed solution returned in the array B and x is the corresponding column of the exact solution X. If RCOND is less than machine precision, then ERRBND is returned as unity.

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL<0 and IFAIL≠-999

If IFAIL=-i, the ith argument had an illegal value.

IFAIL=-999

Allocation of memory failed. The
complex
allocatable memory required is 2×N. In this case the factorization and the solution X have been computed, but RCOND and ERRBND have not been computed.

IFAIL>0 and IFAIL≤N

If IFAIL=i, uii is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

IFAIL=N+1

RCOND is less than machine precision, so that the matrix A is numerically singular. A solution to the equations AX=B has nevertheless been computed.

7 Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form

A+Ex^=b,

where

E1=OεA1

and ε is the machine precision. An approximate error bound for the computed solution is given by

x^-x1x1≤κAE1A1,

where κA=A-11A1, the condition number of A with respect to the solution of the linear equations. F04CCF uses the approximation E1=εA1 to estimate ERRBND. See Section 4.4 of Anderson et al. (1999)
for further details.

8 Further Comments

The total number of floating point operations required to solve the equations AX=B is proportional to nr. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.